Buoyancy-driven convection in Julia-fractal cavities with iteration-dependent boundary evolution

This study investigates buoyancy-driven convection in cavities with Julia-type boundaries, with particular emphasis on Siegel disk configurations. The boundary geometry progressively evolves with the Julia iteration level n , enabling systematic investigation of iteration-dependent boundary evolution and its influence on flow organization and thermal transport. The governing equations, including the incompressible Navier–Stokes and energy equations under the Boussinesq approximation, are solved using a finite-element framework for Rayleigh numbers 10 3 ≤ Ra ≤ 10 6 and Prandtl numbers 0.01 ≤ Pr ≤ 10 . The results show that increasing n progressively modifies the convective response by altering the flow structures, temperature fields, and global transport characteristics. Higher Julia iteration levels produce weaker large-scale circulation and modify the associated heat-transfer response as the cavity boundary becomes progressively refined. The influence of Rayleigh and Prandtl numbers is further examined, revealing enhanced thermal transport with increasing buoyancy forcing and distinct responses of flow intensity and heat-transfer characteristics to variations in transport properties. In addition, a strong linear correlation between horizontal and vertical directional heat-transfer measures is identified, demonstrating the coupling between different components of thermal transport across various operating conditions. Power-law relationships between global flow and heat-transfer quantities further quantify convection scaling in Julia-type cavities. These findings provide insight into how iteration-dependent boundary evolution influences convective transport and geometry–transport interactions in confined thermal systems.